Where is the incenter?
- I was trying to prove for myself the assertion that the locus
of the incenter over all triangles with given Euler segment is
the interior of the orthocentroidal circle, or "arena". However,
the algebra turns out to be a bit too tough even for me - I can
prove that the incenter is always inside this circle, but not that
all points of it are reached.
So I hope that one of the hyacinths (Paul perhaps?) can help by
finding the expression for the squared distance from the incenter
to the midpoint of GH as a function of a,b,c.
I'm also interested in the locus of the excenters. I thought
at first that they would all lie outside the arena, but seem to
have disproved this. There are two problems:
1) what's the set of points where any excenter can be?
2) supposing a > b > c, what are the individual loci of
the a-, b-, and c- excenters?
I suspect that the answer to 1) has a fairly simple shape,
while those that arise in 2) might be quite complicated.
Regards to all, John Conway
>It seems vectors can do the jog without too much trouble (but I confess I
> So I hope that one of the hyacinths (Paul perhaps?) can help by
> finding the expression for the squared distance from the incenter
> to the midpoint of GH as a function of a,b,c.
did not do the dirty work to the last line):
square of IM = inner product of ( (g+h)/2-i) by itself,
where g=(a+b+c)/3 etc all known. Save work by taking a=0 (where of course
I have changed notation to vectors, easily going back to lengths)